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s1. Let A be a UFD and x, y E A two elements having no commor factor; write I by (a, b) (x, y) C A. Prove that y: A? H ax + by is surjective and has kernel the submodule generated by the element (-y, x). In other words, there is an I definec %3D exact sequence (-y,7). (),1-0. 0 A A? This is called the Koszul complex of the pair (x, y). It can obviously happen that I = (x, y) # (1) (for example, take X, Y E A = k[X,Y]). Then I needs two generatoro not a free modulo

s1. Let A be a UFD and x, y E A two elements having no commor factor; write I by (a, b) (x, y) C A. Prove that y: A? H ax + by is surjective and has kernel the submodule generated by the element (-y, x). In other words, there is an I definec %3D exact sequence (-y,7). (),1-0. 0 A A? This is called the Koszul complex of the pair (x, y). It can obviously happen that I = (x, y) # (1) (for example, take X, Y E A = k[X,Y]). Then I needs two generatoro not a free modulo

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. 1. Let A be a UFD and x, y E A two elements having no common factor; write I = (x,y) C A. Prove that p: A? - I defined by (a, b) + ax + by is surjective and has kernel the submodule generated by the element (-y, x). In other words, there is an %3D TI exact sequence 0 → A y,2), 42 (-y,x) (5) →0. This is called the Koszul complex of the pair (x, y). It can obviously happen that I = (x, y) # (1) (for example, take X, Y E A = k[X,Y]). Then I needs two generators, and is %3D not a free module.
Transcribed Image Text:1. 1. Let A be a UFD and x, y E A two elements having no common factor; write I = (x,y) C A. Prove that p: A? - I defined by (a, b) + ax + by is surjective and has kernel the submodule generated by the element (-y, x). In other words, there is an %3D TI exact sequence 0 → A y,2), 42 (-y,x) (5) →0. This is called the Koszul complex of the pair (x, y). It can obviously happen that I = (x, y) # (1) (for example, take X, Y E A = k[X,Y]). Then I needs two generators, and is %3D not a free module.
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